I don't understand how a vacuum, the absence of matter, can hold energy. How can it hold energy when Einstein proved that matter is energy?
And a second related question; how does the energy in a vacuum allow the Universe to produce itself? And how does that account for why the Universe is expanding?
I saw Krauss' lecture on YouTube called Universe from Nothing and only understood some of it, even though he was fantastic at putting things in layman's terms and making these concepts easy to visualize. However, he had a bunch of "asides", so it was a bit confusing. He mentioned that the curvature of the Universe was flat, but then also went on to say how that was the worst possible scenario. Why is a flat Universe such a bad thing?
Best Answer
First of all, Lawrence Krauss has nothing to do with the discovery of dark energy, cosmological constant, or any important discovery, for that matter. This doesn't prove that his ambiguous formulations to suggest otherwise were deliberately written as ambiguous - but it doesn't disprove it, either.
The cosmological constant was introduced by Einstein right after 1916 in an incorrect attempt to make the Universe static and it was observed - at a lower value - in the 1990s experimentally by experimenters which couldn't include Lawrence Krauss. A rather detailed history, explanations, and names of the discoverers of these advances may be found in the very book by Brian Greene, The Hidden Reality, that my Dopplegänger Sheldon Cooper downplayed in the video offered by Raskolonikov, and that I am just translating, in a parallel universe, to my native tongue. ;-)
Second, Einstein's $E=mc^2$ is not an equivalence of energy and matter. It is the equivalence of energy and mass (i.e. the number of kilograms). So a unit volume of the empty space carries some mass equivalent to the energy - it's a mass of a few protons per cubic meter. But the $E=mc^2$ equation does not imply, in any sense, that the mass equivalent to the energy has to take the form of localized particles. It may be dispersed, much like the cosmological constant - whose generalized form is also called dark energy. The main reason why the vacuum contains mass is that this mass contributes to the curvature of spacetime - the gravitational field of mass - and be sure that dark energy does. That is why it was introduced.
Dark energy, unlike mass, carries a negative pressure, and it's the real source of the accelerating expansion it induces. Ordinary matter has attractive gravity.
Third, physics can't label things by moral evaluations such as "worst possible" etc. Krauss probably meant that if the spatial slices of our spacetime are exactly flat - which is the boundary scenario in between the positive curvature and negative curvature - then we have no chance to ever experimentally find out what the magnitude of the curvature is because the sign may always be 0 or $\pm \epsilon$ where $\epsilon$ is sufficiently small relatively to the experimental resolution.
In some sense, the exact flat slices may also be viewed as an example of fine-tuning - something would be zero without any good reason. Some advanced cosmological considerations could imply that the spatial curvature is positive (we can swim around the Universe in principle, after some time), or negative (curved like the Lobachevsky plane, a saddle if you wish).